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A finite family of finite sets $F$

Source: 2016 239 S6

October 11, 2020
combinatorics

Problem Statement

A finite family of finite sets FF is given, satisfying two conditions: (i) if A,B∈FA, B \in F, then A∪B∈FA \cup B \in F; (ii) if A∈FA \in F, then the number of elements ∣A∣| A | is not a multiple of 33. Prove that you can specify at most two elements so that every set of the family FF contains at least one of them.
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